Iterative Concave Rank Approximation for Recovering Low-Rank Matrices

Abstract

In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values, and then minimize the resulting approximation subject to the linear constraints. The accuracy of the approximation is controlled via a scaling parameter δ, where a smaller δ corresponds to a more accurate fitting. The consequent optimization problem for any finite δ is nonconvex. Therefore, to decrease the risk of ending up in local minima, a series of optimizations is performed, starting with optimizing a rough approximation (a large δ) and followed by successively optimizing finer approximations of the rank with smaller δ's. To solve the optimization problem for any δ &gt; 0, it is converted to a new program in which the cost is a function of two auxiliary positive semidefinite variables. The paper shows that this new program is concave and applies a majorize-minimize technique to solve it which, in turn, leads to a few convex optimization iterations. This optimization scheme is also equivalent to a reweighted Nuclear Norm Minimization (NNM). For any δ &gt; 0, we derive a necessary and sufficient condition for the exact recovery which are weaker than those corresponding to NNM. On the numerical side, the proposed algorithm is compared to NNM and a reweighted NNM in solving affine rank minimization and matrix completion problems showing its considerable and consistent superiority in terms of success rate.<p>QC 20150417</p

A High-Resolution DOA Estimation Method With a Family of Nonconvex Penalties

“…This is because the constraints and the derivatives of the objective functions in the two models are the same when the MM method converges. Further, since the objective function in (15) monotonically decreases [36], we can confirm that u * is a local minimum of (15). This completes the proof.…”

A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering

“…However, in the underdetermined setting of CS, it is not possible to have a convex program if sparsity is promoted by a nonconvex penalty. As a result, in [22], [23], the authors propose to use a continuation approach to decline the risk of getting trapped in a local solution without providing any theoretical guarantee. Here, we derive a condition for strict convexity, and this strict convexity allows us to guarantee convergence to the unique optimal solution.…”

“…The performance gap can be seen, for example, in the bias, support recovery, and estimation error [27]- [30]. On the other hand, a number of theoretical and experimental results in compressive sensing and low-rank matrix recovery (LMR) frameworks suggests that better approximations of the 0 norm and the matrix rank result in better performances [22], [23], [28], [31]- [35]. These studies inspire the same result in the mean filtering problem.…”

“…The main idea of our algorithm is to exploit a class of nonconvex functions to approximate the 0 norm more accurately than that of the 1 norm. We use the class of nonconvex penalties introduced in [22], [23] for CS and LMR problems. Nevertheless, contrary to [22], [23], the resulting optimization problem is convex.…”

“…We use the class of nonconvex penalties introduced in [22], [23] for CS and LMR problems. Nevertheless, contrary to [22], [23], the resulting optimization problem is convex. In fact, although the penalty function is nonconvex, since it has a scaling parameter that controls the degree of nonconvexity and is put beside the strictly convex term, y−x 2 , it is possible to keep the optimization problem convex.…”